Using Copula Method and
Changepoint Approach to Survival Data
Bo Qian
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
Credit Risk Modeling and Analysis
Using Copula Method and
Changepoint Approach to Survival Data
Bo Qian
This thesis consists of two parts. The first part uses Gaussian Copula and
Student’s t Copula as the main tools to model the credit risk in securitizations
and re-securitizations. The second part proposes a statistical procedure to identify
changepoints in Cox model of survival data.
The recent 2007-2009 financial crisis has been regarded as the worst financial
crisis since the Great Depression by leading economists. The securitization sector
took a lot of blame for the crisis because of the connection of the securitized products
created from mortgages to the collapse of the housing market. The first part of
this thesis explores the relationship between securitized mortgage products and the
2007-2009 financial crisis using the Copula method as the main tool. We show
in this part how loss distributions of securitizations and re-securitizations can be
On the other hand, the lag effect and saturation effect problems are common
and important problems in survival data analysis. They belong to a general class
of problems where the treatment effect takes occasional jumps instead of staying
constant throughout time. Therefore, they are essentially the changepoint
prob-lems in statistics. The second part of this thesis focuses on extending Lai and
Xing’s recent work in changepoint modeling, which was developed under a time
series and Bayesian setup, to the lag effect problems in survival data. A general
changepoint approach for Cox model is developed. Simulations and real data
anal-yses are conducted to illustrate the effectiveness of the procedure and how it should
List of Tables iv
List of Figures vi
Acknowledgments vii
Chapter 1 Introduction 1
1.1 Credit Risk Modeling and Analysis Using Copula Method . . . 1
1.2 Changepoint Approach to Survival Data . . . 3
Chapter 2 Credit Risk Modeling and Analysis Using Copula Method 5 2.1 Introduction . . . 5
2.1.1 Securitization and Re-securitization . . . 8
2.1.2 Economic Capital and Loss Distribution . . . 9
2.1.3 Valuation vs. Credit Models . . . 13
2.2 Literature Review . . . 15
2.3 Credit Risk Model for Re-securitizations . . . 20
2.3.1 Introduction . . . 20
2.3.2 Some Qualitative Observations . . . 22
2.3.3 The Model . . . 25
2.3.4 Application: Are the AAA Ratings Justified? . . . 46
2.3.5 Derivation of Results . . . 54
2.4 Modeling Credit Risk in Securitizations Using Student’s t Copula . 59 2.4.1 A Review of Copula Functions . . . 59
2.4.2 Connection of ASRF to Gaussian Copula . . . 60
2.4.3 Motivation for Using an Alternative Copula . . . 62
2.4.4 The Student’s t Copula . . . 63
2.4.5 Multivariate t Distribution . . . 65
2.4.6 Securitization Model with Student’s t Copula . . . 67
2.4.7 Numerical Results . . . 69
Chapter 3 Changepoint Approach to Survival Data 72 3.1 Introduction . . . 72
3.1.1 Survival Analysis and Survival Data . . . 72
3.1.2 Cox Model and Partial Likelihood . . . 76
3.2 Model Specification . . . 83
3.2.1 Changepoint Framework . . . 83
3.2.2 Cox Model with Changepoints . . . 89
3.2.3 Estimation Procedure . . . 95 3.3 Proof of Consistency . . . 97 3.3.1 Single Changepoint . . . 97 3.4 Simulation Studies . . . 105 3.4.1 Fixed Segmentation . . . 105 3.4.2 One Changepoint . . . 108 3.4.3 Two Changepoints . . . 117 3.4.4 A Study of Robustness . . . 120 3.5 Data Analysis . . . 123 Chapter 4 Discussion 127 4.1 Credit Risk Modeling and Analysis Using Copula Method . . . 127
4.2 Changepoint Approach to Survival Data . . . 128
References 130
2.1 Moody’s recovery rates on corporate bonds, 1982-2003. . . 24
2.2 EL and LCL withK = 30 . . . 43
2.3 CDO Tranche 15% - 50% . . . 44
2.4 EL and UL of RMBS and CDO tranches . . . 50
2.5 EL and UL of RMBS and CDO tranches (continued) . . . 51
2.6 Student’s t Copula with different degrees of freedom . . . 69
3.1 One changepoint with Bernoulli covariate, fixed segmentation. . . . 106
3.2 One changepoint with normal covariate, fixed segmentation. . . 107
3.3 One changepoint with Bernoulli covariate, natural segmentation. . . 109
3.4 One changepoint with normal covariate, natural segmentation. . . . 111
3.5 Varying changepoint locations. . . 112
3.6 Varying degrees of censoring. . . 113
3.7 Varying scales of change . . . 114
3.8 One changepoint with number of changepoint unknown. . . 115
3.11 Two changepoints with number of changepoint known (b). . . 118
3.12 Two changepoints with number of changepoint unknown (a). . . 119
3.13 Two changepoints with number of changepoint unknown (b). . . 120
3.14 Treatment effect continuous change. . . 123
2.1 Securitization and re-securitization . . . 10
2.2 Granularity adjustments . . . 45
2.3 Amount of support needed to make the top RMBS tranche ‘AAA’ . 53 2.4 EL and LCL under different degrees of freedom . . . 70
3.1 Treatment effect steep change . . . 121
3.2 Treatment effect gradual change . . . 122
3.3 Kaplan-Meier estimates of survival probabilities . . . 124
3.4 Changepoint procedure estimate . . . 125
3.5 MLE procedure estimate . . . 126
I would like to express my deepest gratitude toward my Ph.D advisor, Professor
Zhiliang Ying, from whom I have received so much advice, constant support, and
constructive criticism on research. I would like to thank Professor Haipeng Xing for
his kind help on the changepoint model, and thank Professor Tian Zheng, Victor
de la Pena, Quanshui Zhao, and Yixin Fang for their insightful comments.
In addition, I am deeply thankful to my parents, whose love has supported me
during my life in the States. I would like to give special thanks to my wife,
Yi-Hsuan Lee, without whose patience and encouragement I would not have been able
to complete this thesis.
Bo Qian
May 2013
Chapter 1
Introduction
This thesis consists of two parts. The first part utilizes Gaussian Copula and
Student’s t Copula to model the credit risk in securitizations and re-securitizations.
The second part proposes a statistical procedure to identify changepoints in Cox
model of survival data.
1.1
Credit Risk Modeling and Analysis Using
Cop-ula Method
The financial crisis of 2007-2009 has been regarded as the worst financial crisis
since the Great Depression by leading economists. During a two-year span, we
have witnessed the whole US financial system and all the institutions going through
failures from 2005 to 2007. The crisis also saw millions of home owners losing their
houses to foreclosures. The majority of these mortgages are so-called subprime or
Alt-A, since the borrowers tend to be of lesser credit worthiness as opposed to prime
borrowers. Therefore, the 2007-2009 crisis is often referred to as the ‘subprime crisis’
as well.
The securitization sector in the financial industry took a lot of blame during and
after the subprime crisis and credit crunch. This sector is very closely connected to
mortgages because of the enormous number of securitizations and re-securitizations
created from residential mortgages before and during the crisis. These securities,
often referred to as residential mortgage-backed securities (RMBS) and
collateral-ized debt obligations (CDO) backed by RMBS tranches, were in the center of this
turmoil and were direct or indirect causes of many of the bank failures. They relate
to plenty of factors that are generally deemed to have contributed to the crisis, e.g.,
subprime lending, housing bubble, complexity and lack of transparency of financial
products, incompetence of credit ratings, misuse or misunderstanding of profound
mathematical models, etc.
The first part of this thesis is largely an effort to explore the relationship between
these securitized products and the subprime crisis using Copula method as the
main tool, attempting to answer the questions of what kind of role they played
in the crisis. We show in this part how loss distributions of securitizations and
the model is also used to examine whether and where the ratings of securitized
products could be flawed.
1.2
Changepoint Approach to Survival Data
Cox model is the most widely used regression model in survival data analysis.
It models failure time by decomposing the hazard rate function into two parts:
the baseline hazard function, describing how hazard (risk) changes over time at
‘baseline’ levels of covariates; and the effect parameters, describing how the hazard
varies in response to explanatory covariates.
In survival analysis, it is often not unreasonable to assume that a treatment
does not start to affect the risk of failure until after a certain time lag, which can
be called “lag effect”. There is also the possibility of a “saturation effect”, where
the treatment stops to affect the risk of failure after a certain period of time. In
the terms of Cox model, these situations are equivalent to a time-varying covariate
effect parameter, staying at a constant value for a period of time and then taking a
jump to a new value. The statistical problem of identifying the location of jumps
falls into a general class of problems called changepoint problems.
Changepoint problems have been studied in the statistical literature since the
1970s, and most recently by Lai and Xing (2011). The second part of this thesis
data and develops a changepoint procedure for Cox model. Simulations and real
data analyses are conducted to illustrate the effectiveness of the procedure and how
Chapter 2
Credit Risk Modeling and
Analysis Using Copula Method
2.1
Introduction
The financial crisis of 2007-2009 has been regarded as the worst financial crisis
since the Great Depression by some leading economists. During a two-year span,
we have witnessed the whole US financial system and all the institutions going
through extraordinary changes.
In March 2008, the concerns that investment bank Bear Stearns would collapse
resulted in its fire-sale to JP Morgan Chase. The crisis hit its peak in September
and October 2008. Several major institutions either failed, were acquired under
Merrill Lynch, Fannie Mae, Freddie Mac, and AIG. The receivership of Washington
Mutual Bank by federal regulators on September 26, 2008, was the largest bank
failure in U.S. history. Twenty-five banks in the United States failed and were
taken over by the Federal Deposit Insurance Corporation (FDIC) in 2008, after
only three failures in 2007 and none in 2006 or 2005. According to CNN, 140 more
banks failed in 2009, causing the FDIC’s deposit insurance fund to slip into the
red for the first time since 1991. In a dramatic meeting on September 18, 2008,
Treasury Secretary Henry Paulson and Fed Chairman Ben Bernanke met with key
legislators to propose a $700 billion emergency bailout.
The US economic and financial system underwent a huge contraction period
during the crisis. According to the Bureau of Economic Analysis, Real Gross
Do-mestic Product, the output of goods and services produced by labor and property
located in the United States, decreased at an annual rate of approximately 6
per-cent in the fourth quarter of 2008 and first quarter of 2009, versus activity in the
year-ago periods. According to the Bureau of Labor Statistics, the U.S.
unemploy-ment rate increased to 9.8% by September 2009, the highest rate since 1983 and
roughly twice the pre-crisis rate. The average hours per work week declined to 33,
the lowest level since the government began collecting the data in 1964.
Roger Altman (2009), who was U.S. Deputy Treasury Secretary in 1993-94,
wrote that “the crash of 2008 has inflicted profound damage on the US
geopolitical setback...the crisis has coincided with historical forces that were already
shifting the world’s focus away from the United States. Over the medium term, the
United States will have to operate from a smaller global platform—while others,
especially China, will have a chance to rise faster.”
Since the start of the crisis, economists and financial researchers have been
re-flecting and contemplating, trying to figure out the real cause of the crisis behind
the scenes. Most people tend to agree that the dramatic rise in mortgage
delin-quencies and foreclosures was the fuse that set off most of the events afterwards.
That is why this past crisis is also referred to as the ‘subprime crisis’, since most
of the delinquencies came from subprime and Alt-A mortgages, which are typically
of lesser credit worthiness as opposed to prime mortgages.
The securitization sector in the financial industry took a lot of blame during the
subprime crisis and credit crunch. This sector is very closely connected to mortgages
because of the enormous number of securitizations and re-securitizations created
from residential mortgages before and during the crisis. These securities, often
re-ferred to as residential mortgage-backed securities (RMBS) and collateralized debt
obligations (CDO) backed by RMBS tranches, were in the center of this turmoil
and were direct or indirect causes of many of the bank failures. They relate to
plenty of factors that generally deemed to have contributed to the crisis, e.g.,
sub-prime lending, housing bubble, complexity and lack of transparency of financial
mathematical models, etc.
This part of the thesis is largely an effort to explore the relationship between
these securitized products and the subprime crisis through a mathematical way,
attempting to answer the questions of what kind of role they played in this crisis.
2.1.1
Securitization and Re-securitization
The financial product of main interest in this thesis will be the so-called
‘struc-tured products’, namely ‘securitizations’ and ‘re-securitizations’. They got this
name because they are usually structured and ‘tranched’ in such a way that will
give investors different choices for risk and return profile.
As illustrated by Figure 2.1, the typical practice of structuring a re-securitization
deal is as follows: firstly, thousands of mortgages (often times of subprime quality)
are gathered into one portfolio and securitized. The deal is sliced into multiple
tranches which, with different risk and return profile, would cater to the needs of
different investors. The top tranche (usually calledsuper-senior orsenior tranche), which is of the highest quality and lowest risk, will often be bought by big
insti-tutions like insurance companies. The middle tranches (usually called mezzanine
tranches) are more risky but generate bigger returns. The lowest tranche is usually
called ‘equity’ tranche for the similarity of its position in a liability structure to a
stock. Equity tranches are very risky and are usually sold to hedge funds that seek
‘RMBS’ in this case.
The second step, called ‘re-securitization’, involves securitizing tens or
hun-dreds of these RMBS tranches (usually mezzanine tranches or subordinate senior
tranches) again. This portfolio of RMBS tranches is also sliced into senior,
mez-zanine and equity tranches and sold to various investors. This kind of product is
called a ‘CDO of RMBS’.
There is about $14.6 trillion in total U.S. mortgage debt outstanding. There
are about $8.9 trillion in total U.S. mortgage-related securities. Mortgage-backed
securities can be considered to have been in the tens of trillions, if ‘Credit
De-fault Swaps’ (CDSs) are taken into account. (‘Credit DeDe-fault Swaps’ are synthetic
instruments that reference a tranche of a securitization or re-securitization.)
In general, securitizations and re-securitizations do not have to be structured
from mortgages only. As a matter of fact, people construct them from student
loans, car loans, corporate loans, etc. in practice. These products may differ in
certain deal specifics from mortgage securities, but the overall ideas of tranching
and risk/return tradeoff remain exactly the same.
2.1.2
Economic Capital and Loss Distribution
In finance, mainly for banks and other financial services firms, economic capital
is the amount of risk capital, assessed on a realistic basis, which a firm requires
Figure 2.1: Securitization and re-securitization
CDO of RMBS
Re-Securitization Securitization AAA AAA AAAmarket risk, credit risk, and operational risk. It is the amount of money which
is needed to secure survival in a worst case scenario. Firms and financial services
regulators should then aim to hold risk capital of an amount equal to or above
economic capital.
Dev (2004) introduced the concept of economic capital as follows:
“Economic capital is a measure of risk. It is a single measure that captures the
unexpected losses and reduction in value or income from a portfolio or business
in a financial institution. The risk arises from the unexpected nature of the losses as distinct from expected losses, which are considered part of doing business and are covered by reserves and income. Economic capital covers all unexpected events
except the catastrophic ones, for which it is impossible to hold capital. Economic
capital is a common currency in which all risks of a financial institution can be
mea-sured, enabling comparison of risk across different risks, across diverse businesses
and across different financial institutions.”
Formally, economic capital is often calculated by determining the amount of
money that the firm needs to ensure that its balance sheet stays solvent over a
cer-tain time period (i.e., the time period is usually one year if not specified otherwise)
with a pre-specified probability. Therefore, economic capital is often calculated as
‘value at risk’. In the area of credit risk, economic capital or simply capital, is
usually calculated from the quantiles of the loss distribution of an instrument or a
The concept of economic capital differs from regulatory capital in the sense that
regulatory capital is the mandatory capital the regulators require to be maintained
while economic capital is the best estimate of required capital that financial
institu-tions use internally to manage their own risk and to allocate the cost of maintaining
regulatory capital among different units within the organizations.
This thesis will mainly deal with the problem of how to calculate capital for
securitizations and re-securitizations. There have been a lot of studies both in
academics and in industry on the loss distribution of a credit portfolio and on the
loss distribution of a securitization tranche supported by a credit portfolio, since
securitizations like RMBS have existed in the U.S. since 1970. However, a lot of
re-securitization deals came onto the scene only after 2000. They are more complicated
deals than securitizations, and they are quite different from securitizations in a few
very important aspects. Regrettably, a lot of people in the industry did not realize
or put enough emphasis on those differences, which leads to them buying a lot
of re-securitization products (e.g., CDO tranches) without fully understanding the
risk that accompanies these products.
What compounded the problem was, at least as it appeared throughout the
crisis, that the rating agencies that rate these products (e.g., S&P, Moody’s, Fitch)
did not model the re-securitization products accurately enough. However, a lot of
investors were very much dependent on these agency ratings. Since they do not
judge the riskiness of these products. A lot of them were under the ‘illusion’ that
a corporate bond, an RMBS tranche and a CDO tranche have exactly the same
risk if they have the same ratings. Of course, as it turned out, this is totally a
misconception if not more. See, for example, Hull and White (2010).
On the other hand, the Basel II accord, which most regulatory capital
requiments are based upon in the U.S. and Europe, did not differentiate capital
re-quirements of securitizations from those of re-securitizations until recently. See
Basel Committee on Banking Supervision (2009) for more details. Additionally,
the capital requirements in Basel II are largely tied to agency ratings as well.
Ret-rospectively, the inadequate amount of capital kept by some financial institutions
contributed to their downfall as well.
2.1.3
Valuation vs. Credit Models
The typical ‘wall-street’ approach toward the structured products has been by
copula-based simulations, which could capture to some degree the correlation
be-tween different obligors and the complicated priority of payments (also called
‘wa-terfall’) in those securitization products. This approach is relatively accurate
be-cause of the incorporation of deal by deal specifics, but as most Monte Carlo
sim-ulation methods it is rather time-consuming. These products were very actively
traded before the crisis, so the main objective of the people running this type of
appro-priate for this purpose to calibrate the model to market prices down to nickels and
dimes. Refer to Li (2000) and Hull and White (2010) for discussions of copula-based
simulation models.
However, for the purpose of setting capital requirements of hundreds or
thou-sands of securitization and re-securitization tranches that a large bank or
insur-ance company hold, the simulation approach is not ideal because of its
time-consumingness. One is more concerned about the possible extreme credit losses
on the portfolio than whether the portfolio is worth 95 cents on a dollar or 96
cents. It is even a heavier computational burden if one needs to get an accurate
estimate of a high quantile of the loss distribution. That is why a credit model with
a simplified, fast and analytical or at least half analytical approach would be more
desirable. Such a methodology would help setting and benchmarking regulatory
capital standards as well.
Of course, an analytical model will not be able to incorporate details of each
deal’s waterfall, or other factors such as prepayment risk, or interest rate term
structure, but with loss as the main focus, the strengths of such a model still
outweighs the weaknesses.
The above facts clearly suggest the need for an analytical model that describes
the loss distribution of a re-securitization tranche and from which people can get
quantiles rather efficiently. We already have something to build upon, which is the
(ASRF) framework.
ASRF is a large array of models defined by Gordy (2003), which has desirable
properties in terms of portfolio loss distribution. Pykhtin and Dev (2002b) proposed
a specific model in this category that has close ties with the Gaussian Copula model
in order to calculate the loss distribution and capital of securitizations.
In the first part of this chapter, we will extend the Pykhtin-Dev model to
re-securitizations and show how to calculate analytically the loss distribution and
capital of a re-securitization tranche. In the second part, we will first generalize
the correlation structure in Pykhtin-Dev Model from Gaussian Copula to Student’s
t Copula for the purpose of incorporating higher tail risk and tail dependence,
and then propose a semi-analytical approach to calculate the loss distribution and
capital of a securitization tranche in the new model. Both of the new models will
be in the ASRF framework.
2.2
Literature Review
2.2.1
Asymptotic Single Risk Factor Models
In his 2003 foundational paper, Gordy defined a category of portfolio credit
risk models: the Asymptotic Single Risk Factor (ASRF) model. These models are
simple yet powerful tools for analyzing credit risk in portfolios and securitizations.
Pykhtin-Dev model proposed in Pykhtin and Pykhtin-Dev (2002b), which the Basel II rating-based
approach of capital requirement is based on.
The ASRF model is attractive mainly because of its very desirable asymptotic
properties. Some of the important properties are cited below from Gordy (2003)
to help develop results in the following sections.
For a portfolio of n obligors, define the portfolio loss ratio Ln as the ratio of
total losses to total portfolio exposure, i.e.,
Ln ≡ n ∑ i=1 UiAi n ∑ i=1 Ai , (2.1)
whereAi is the exposure to obligor i, which is known and non-stochastic;Ui is the
random variable that denotes loss per dollar exposure. In the event of survival,
Ui = 0. Otherwise, Ui is the percentage loss-given-default (LGD) on instrument i.
For a given q∈(0,1), value-at-risk (VaR) is defined as the qth quantile of the loss distribution, and is denoted VaRq[Ln]. Let αq(Y) denote the qth quantile of the
distribution of random variable Y, hence, VaRq[Ln] =αq(Ln).
LetX denote the systematic risk factors of the portfolio, which are drawn from
a known joint distribution. It is assumed that all dependence across credit events is
due to common sensitivity to these factors. Conditional onX, the remaining credit
risk of the portfolio is idiosyncratic to the individual obligors in the portfolio.
In general, X can be multi-dimensional or uni-dimensional. Multi-dimensional
dimensions of systematic factors in a (mortgage) portfolio. Geography, maturity,
and fixed rate vs. adjustable rate are all possible dimensions. However, a
uni-dimensional systematic factor offers a great advantage in calculating the loss
dis-tribution and capital, as we will see later. Even very simplistic multi risk factor
models are usually reliant on computationally intensive simulation methods, while
single risk factor models can have elegant analytical solutions. There are
method-ologies that approximate a multi-risk factor model by a single risk factor model (and
an adjustment term), which in essence resembles the principal component method.
Please refer to Pykhtin (2004) for an introduction to multi-risk factor models. In
this thesis, we will focus on single risk factor models.
Formally, we will assume that
(A-1) the {Ui} are bounded in the interval [−1,1] and, conditional on X, are
mutually independent.
(A-2) the Ai are a sequence of positive constants such that
(a) ∑ni=1Ai ↑ ∞ and
(b) there exists a ζ >0 such that An/
∑n
i=1Ai =O(n−(1/2+ζ)).
The restrictions in (A-2) are sufficient to guarantee that the share of the largest
single exposure in total portfolio exposure vanishes to zero as the number of
weak and would be satisfied by any conceivable real-world portfolio. Gordy (2003)
proved the following theorem.
Theorem 1. If (A-1) and (A-2) hold, then, conditional onX =x,Ln−E[Ln|x]→
0, almost surely.
In intuitive terms, Theorem 1 says that as the exposure share of each asset
in the portfolio goes to zero, idiosyncratic risk in portfolio loss is diversified away
perfectly. In the limit, the loss ratio converges to a fixed function of the
system-atic factor X. We refer to this limiting portfolio as “infinitely fine-grained” or as
an “asymptotic portfolio.” An implication is that, in the limit, we only need to
know the conditional distribution of E[Ln|X] to answer any questions about the
unconditional distribution of Ln.
Let Fn denote the cumulative distribution function (cdf) of Ln. The second
theorem proved by Gordy (2003) can be described as follows.
Theorem 2. If (A-1) and (A-2) hold, then for any ϵ >0,
Fn ( αq(E[Ln|X]) +ϵ ) →[q,1], Fn ( αq(E[Ln|X])−ϵ ) →[0, q]. (2.2)
The importance of Theorem 2 is that it allows us to substitue the quantiles of
E[Ln|X] (which often times are easier to calculate) for the corresponding quantiles
of the loss ratioLn(which are usually difficult to calculate) as the portfolio becomes
Furthermore, the quantiles ofE[Ln|X] take on a particularly simple and
desir-able asymptotic form when we impose two additional restrictions:
(A-3) the systematic risk factor X is one-dimensional; and
(A-4) E[Ui|x] are nondecreasing inx for all i.
If we define Mn(x)≡E[Ln|x] = n ∑ i=1 E[Ui|x]Ai n ∑ i=1 Ai , (2.3)
the third theorem proven in Gordy (2003) is stated as follows.
Theorem 3. If (A-3) and (A-4) are satisfied, then αq(E[Ln|X]) =E[Ln|αq(X)] =
Mn(αq(X)).
The importance of this result lies in the linearity of the expectation operator.
Whereasαq(E[Ln|X]) may, in the general case, be highly complicated,E[Ln|αq(X)]
is simply the exposure-weighted average of the individual assets’ conditional
ex-pected losses. Taken together with Theorems 1 and 2, Theorem 3 permits a simple
and powerful rule for determining capital requirements. For asset i, allocate
cap-ital per dollar book value (inclusive of expected loss) of ci ≡ E[Ui|αq(X)] +ϵ, for
some arbitrarily small ϵ. Observe that this capital charge depends only on the
characteristics of instrument i and thus this rule is portfolio-invariant.
Models that satisfy all the assumptions (A-1) to (A-4), thus having the
Factor (ASRF) model.
Please refer to Gordy (2003) for proofs and details of the Theorems.
2.2.2
Pykhtin-Dev Model
Pykhtin and Dev (2002a) developed an analytical model to calculate the loss
distribution and economic capital of securitizations under the ASRF framework.
Because of the close connection of their model to our main model of this chapter,
The detailed discussion of the Pykhtin-Dev model will be in Section 2.3.3.1.
2.3
Credit Risk Model for Re-securitizations
2.3.1
Introduction
In the financial turmoil that started in the latter half of 2007, clearly a
dis-proportionate role has been played by the billions of dollars of marked-to-market
losses suffered by financial institutions on Collateralized Debt Obligations (CDOs)
created out of tranches of Residential Mortgage-backed Securities (RMBS), which
in turn were created from subprime and Alt-A mortgages. Such CDOs may be
cash CDOs or synthetic CDOs (Credit Default Swaps written with a super-senior
tranche of a CDO of RMBS as reference entity).
CDO of RMBS is a special case of recent types of structured credit products
the fact that a re-securitization like a CDO of RMBS is more complex and less
transparent than a securitization like RMBS. Yet the average investor has also
been content with the rating. After all, both the senior-most tranches of a CDO of
RMBS and of a RMBS are rated AAA.
But it is not enough to talk about complexity and transparency in order to have
a good understanding of the relative risks of an investment in AAA CDO of RMBS
tranche and of an investment in AAA RMBS tranche. The fact that both of them
are AAA-rated seems to indicate, at a first glance, that the credit risk in the two
types of securities must be rather similar; especially if the structures (securitization
and re-securitization) are created out of the same pool of underlying mortgages. In
this paper, we show that this statement is not necessarily true.
It has progressively become clear over time that there has been rather an
al-most complete lack of understanding of the credit risk in re-securitizations such
as CDO of RMBS, particularly tail risk measures. The Basel Committee has
re-cently published a proposal revamping the capital adequacy rules for
securitiza-tion. The proposal contains different sets of capital factors for securitization and
re-securitization (BCBS, 2009), making a clear distinction between securitization and re-securitization.
Yet there is very little in the published credit risk literature that models credit
risk in re-securitization explicitly and that is distinct from credit risk in
by the rating agencies have been mostly ad-hoc and in any case do not address this
basic distinction.
In this section we develop a complete model for credit risk in re-securitizations.
The model applies to all re-securitizations which are created from primitive
as-sets that are highly diversified. However, for presentation conciseness, we focus
specifically on mortgage-related re-securitizations namely, CDO of RMBS.
The main objective of this section is to present a model for credit risk in
re-securitizations. The numerical results are for illustration only. The numerical
results from the model show that, starting from the same pool of mortgages, a
super-senior RMBS tranche rated AAA and a super-senior CDO tranche rated
AAA actually have very different credit risk characteristics.
2.3.2
Some Qualitative Observations
We can make several qualitative observations on the difference between a
secu-ritization tranche and a re-secusecu-ritization tranche.
The fundamental difference lies in the fact that they have collateral support that
have very different characteristics. For securitization tranches, usually the collateral
consists of stand-alone bonds, loans, or in the case of RMBS, mortgages; while for
re-securitization tranches, the collateral consists of RMBS tranches or tranches of
other Asset-Backed Securities (ABS).
loss-given-default, a mortgage rarely has a 100% loss when it defaults. Even in a hostile
environment like recently when housing prices have decreased a lot, LGD on a
mortgage is usually between 50% and 60% at worst. Other asset-backed securities
share similar collateral LGD levels as well. For example, Table 2.1 shows the
Moody’s historical recovery rates on corporate bonds. Recovery rate is defined as
the complement of loss-given-default. One can see that senior secured bonds, which
have the highest priority among all corporate bonds, exhibit an average recovery
rate of above 50%. Meanwhile, the unsecured or subordinate bonds have average
recovery around 30%. However, thin RMBS tranches tend to have very high loss
levels when they default, which conceptually can be understood easily, since once
the tranche is hit, it only takes a few more defaults before it is completely gone.
It turns out that the mezzanine RMBS tranches that people use most to construct
CDO deals are usually very thin (around 1%), which leads to very high LGD’s.
For the same reason above, one can also say that the loss distributions of a
mort-gage and an RMBS tranche are very different. The support of the loss distribution
of a mortgage rarely includes the 100% point. However, the loss distribution of an
mezzanine RMBS tranche will have point masses on the 0% point and the 100%
point, since it could incur no loss at all or it could lose every penny of its principal.
Secondly, individual mortgages have relatively low correlation between each
other, because of high idiosyncracy. The Basel II standard for correlation
Table 2.1: Moody’s recovery rates on corporate bonds, 1982-2003.
Class Average recovery rate (%)
Senior secured 51.6
Senior unsecured 36.1
Senior subordinated 32.5
Subordinated 31.1
Junior subordinated 24.5
Note. The recovery rates are expressed as a percentage of face value.
between each other, for the fact that through the securitization process,
idiosyncra-cies among different mortgages are largely diversified away. As Gordy’s results have
shown, the risk left is mainly systematic risk. Although different RMBS tranches
do not necessarily share the same systematic risk, but they will tend to be much
more highly correlated.
The third point is relatively subtle. There have been quite some debates on
whether defaults and LGD’s are correlated for (corporate) bonds, with the
ma-jority of people leaning toward the positive answer. Efforts have been made to
incorporate such correlations in the determination of capital, e.g., by the
intro-duction of ‘Downturn LGD’. Refer to, for example, Miu and Ozdemir (2006) for a
detailed discussion on Downturn LGD. This type of correlation also applies to the
case of RMBS tranches, but in a more evident way. For instance, when housing
prices decline, not only will the defaults on RMBS tranches go up, but the LGD
in the current crisis.
Higher LGD and higher correlation as stated above make the loss distribution
of the collateral portfolio of an RMBS and a CDO very different. As a result, the
loss distribution of an RMBS tranche and a CDO tranche will be very different.
The proposed new model will naturally incorporate all these considerations.
On the other hand, a lot of financial institutions and rating agencies did not
recognize the difference between a regular bond and a securitization tranche, which
probably is one of the main reasons why their models failed in this crisis.
2.3.3
The Model
Our objective is to develop a model for credit risk in re-securitization (in
par-ticular CDO of RMBS). In doing so, we adopt a “look-through” approach which
means we start from the credit characteristics of the underlying primitive loans.
In that spirit we first introduce the basic notations for credit risk metrics for the
portfolio of loans (in particular mortgages) and for the securitization tranches (in
particular RMBS) underneath the re-securitization in subsection 2.3.3.1, where we
make use of already existing results in the literature. Discussion of our model really
2.3.3.1 Loss Distribution of RMBS Tranche - Securitization
Consider an RMBS which has underlying collateral of M homogeneous
mort-gages with the same probability of default (PD) p, expected loss-given-default
(LGD) µ, asset-value correlation (AVC) ρ and equal weights. We will focus on
homogeneous portfolios to keep equations and derivations simple. As pointed
out later, our results can be extended to non-homogeneous portfolios in a rather
straightforward way.
Following Pykhtin and Dev (2002a), we have a set of continuous random
vari-ables {Vj}Mj=1 that describe the financial well being of each mortgage. These ran-dom variables have standard normal distribution and trigger defaults whenever
Vj < VjD ≡ N−1(p). N(·) and N−1(·) stand for the cumulative distribution
func-tion (CDF) and the inverse cumulative distribufunc-tion funcfunc-tion of the standard normal
distribution, respectively. They can be written as
Vj =√ρY +
√
1−ρWj, (2.4)
where Y is the systematic risk factor and {Wj}are idiosyncratic factors. All these
factors are independent of each other and have standard normal distribution. We
also assume independent (independent of Y and {Wj} as well) and identically
distributed LGD’s with expected value µ. In general, {Vj} could stand for the
asset value of a firm, or the value of a bond, etc. Then the above equation describes
So this is a very general model that covers other securitizations as well, such as
Collateralized Bond Obligations (CBO), Collateralized Loan Obligations (CLO)
and Asset-Backed Securities (ABS).
Please note that the systematic risk factor in this thesis here and afterwards is
negatively correlated to the loss of the mortgage, as one can easily see from equation
(2.4). This is the most intuitive and widely accepted way of specifying loss and
asset value. So exactly opposite to (A-4), E[Ui|x] will be nonincreasing in x for all
i instead of nondecreasing. But the quantile argument of Theorem 3 still holds,
since one just needs to change the sign of the systematic factor to apply the result.
Or instead of
αq(E[Ln|X]) =Mn(αq(X)), (2.5)
one will have this equation:
αq(E[Ln|X]) =Mn(α1−q(X)). (2.6)
We will make use of this slightly altered Gordy’s result multiple times from now.
However, we will not mention this change of sign every time to avoid repetition.
SupposeM is large enough so that the portfolio can be considered fine-grained.
Because losses in all the mortgages are affected by a single systematic risk factor,
this fits right into the ASRF framework. Thus the loss distribution of the collateral
factor Y: L∞(Y) ≡ E[LColl|Y] = M ∑ j=1 E[LGDj ·1{Vj<N−1(p)}Y] = µ M ∑ j=1 P(√1−ρWj < N−1(p)−√ρY ) = µN ( N−1(p)− √ρY √ 1−ρ ) , (2.7)
where LGDj is the LGD on the jth mortgage and 1{·} is the indicator function.
Thus we have theqth quantile of loss approximately to beL∞(Y1−q) where Y1−q =
N−1(1−q).
We denote by G(l) the probability of loss exceeding l, and by (2.7),
G(l) = N ( 1 √ρ [ N−1(p)−√1−ρN−1 ( l µ )]) for l < µ. (2.8)
Theorem 4(Pykhtin & Dev, 2002b)
For any tranche [t1, t2] such thatt2 < µ, the expected loss (EL) can be computed
by E[LRMBS]= µ t2−t1 N2 ( N−1(p), N−1 ( t µ ) ;√1−ρ) t2 t1 , (2.9)
where N2(a, b;γ) denotes the bivariate normal distribution valued at (a, b) with
mean 0, standard deviation 1 and correlation γ.
For those tranches [t1, t2] such that t1 < µ < t2, equation (2.9) will still hold if
we replace the term N−1(t/µ) by N−1(min (t/µ,1)). In this case,
N2
(
N−1(p),∞;√1−ρ
)
for any t≥µ.
Since the portfolio or the tranche loss distribution can be characterized by the
single systematic factor Y, we also have the following property:
Corollary 1. The qth quantile of the tranche loss or loss at the confidence level
(LCL) is L∞(Y1−q)−t1 t2 −t1 if t1 < L∞(Y1−q)< t2 0 if L∞(Y1−q)< t1 1 otherwise .
One could use this LCL as the capital for the RMBS tranche if looking at the
tranche in a stand-alone fashion. But in reality, investors or banks usually buy this
tranche and put it into their own investment portfolio. This portfolio is usually
very big especially for big banks, and we will refer to this portfolio as the
‘super-portfolio’ in the future to differentiate it from the collateral portfolio. This concept
originates from Pykhtin and Dev (2002b) and is exactly the concept of marginal
VaR in the value-at-risk framework (see, for example, Jorion, 2006). So the real
question of interest is how much capital the bank should allocate when it adds
this RMBS tranche to its super-portfolio. This kind of capital in the context of a
super-portfolio will be our focus here and afterwards.
The concept of capital in the context of a super-portfolio not only is more
(see Pykhtin & Dev, 2002b). As we will see later in the numerical examples, the
stand-alone capital tends to be 100% for junior tranches and 0% for senior tranches.
This phenomenon makes the stand-alone capital very unsatisfactory particularly for
regulatory purpose. (For example, this would mean that literally all ‘AAA’-rated
tranches will be allowed ‘zero’ capital.)
We would like to apply Gordy’s results in the next step. In order to do that,
we need to make some assumptions on the super-porfolio, namely (A-1)−(A-4). In more practical terms, we have assumed that
i. the super-portfolio where the tranche is held is asymptotically fine-grained;
ii. the super-portfolio is driven by another single systematic risk factor Z; and
iii. investors exposure to the tranche is small compared to total exposure in the
super-portfolio.
These are weak enough conditions that most portfolios of big institutions will
sat-isfy. Refer to Pykhtin and Dev (2002b) for details.
Suppose that we place the RMBS tranche into a super-portfolio of which the
systematic factor Z is correlated with Y as
Y =√λZ+√1−λη, (2.10)
where the correlation is denoted byλ and η represents a standard normal random
all the requirements of the ASRF model, then in order to calculate the capital for
the RMBS tranche in the context of the super-portfolio, we only need to apply
Theorem 3 and compute E[LRMBS|Z].
First we could compute the probability of loss exceeding levell given the
super-portfolio systematic factor Z:
G(l|Z) =N ( 1 √ ρ(1−λ) [ N−1(p)−√1−ρN−1 ( l µ ) −√ρλZ ]) for l < µ. (2.11)
Hence for any tranche t2 < µ, the conditional expected loss of the tranche will be
E[LRMBS|Z] = 1 t2−t1 ∫ t2 t1 dlG(l|Z) = µ t2−t1 N2 ( N−1(p)−√ρλZ √ 1−ρλ , N −1 ( t µ ) ; √ 1−ρ 1−ρλ ) t2 t1 . (2.12)
For those tranches [t1, t2] such thatt1 < µ < t2, we could replace the termN−1(t/µ)
byN−1(min (t/µ,1)) so that (2.12) still holds. Note that for any t≥µ,
N2 ( N−1(p)−√ρλZ √ 1−ρλ ,∞; √ 1−ρ 1−ρλ ) =N ( N−1(p)−√ρλZ √ 1−ρλ ) .
Theorem 5(Pykhtin & Dev, 2002b)
The capital of an RMBS tranche [t1, t2] at the (100×q)% confidence level is simply E[LRMBS|Z
1−q
]
as defined in (2.12).
From the above expressions of EL and LCL of an RMBS tranche given in (2.9)
First, for the loss of a whole portfolio, we know that the expected loss does not
depend on anything but PD and LGD. But for the loss of an RMBS tranche, not
only does it depend on these two parameters, it also depends on the asset value
correlation as well. In general, equity tranches’ EL decreases when AVC increases,
while senior tranches’s EL increases when AVC increases. In other words, higher
correlation favors equity tranches, but disfavors senior tranches. There is a grey
area of mezzanine tranches in the middle, whose direction of movement will depend
on their position and other deal specifics.
Second, for LCL or capital in the context of a super-portfolio, correlation to the
super-portfolio plays an important role. It is not very straightforward though to
see how it impacts the capital. Let’s consider extreme situations first:
If λ = 0, it means that the loss of this tranche is independent of the total
performance of the super-portfolio. Plugging λ= 0 into equation (2.12), we get
E[LRMBS|Z] = µ t2 −t1 N2 ( N−1(p), N−1 ( t µ ) ;√1−ρ) t2 t1 =E[LRMBS].
So in this case, LCL will be equal to EL for all tranches, meaning that no matter how
the super-portfolio performs, the conditional expected loss of the RMBS tranche
will always be equal to the unconditional EL.
If λ = 1, it means that the loss of this tranche is perfectly correlated with the
get E[LRMBS|Z] = µ t2−t1 N2 ( N−1(p)− √ρZ √ 1−ρ , N −1 ( t µ ) ; 1) t2 t1 = L∞(Y1−q)−t1 t2−t1 if t1 < L∞(Y1−q)< t2 0 if L∞(Y1−q)< t1 1 otherwise ,
which means that LCL in the context of the super-portfolio will be equal to the
stand-alone capital.
In practice, because of its interpretation as the correlation between two
sys-tematic factors, λ is usually set between 0.5 and 1. In this case, compared to the
stand-alone capital, λ has the effect of transferring capital from junior tranches to
senior tranches. Refer to Pykhtin and Dev (2002b) for more details.
2.3.3.2 Loss Distribution of CDO Tranche - Re-securitization
Now we consider a CDO comprised ofKhomogeneous RMBS mezzanine tranches
[t1, t2] taken from different mortgage pools. Suppose the loss of the collateral (i.e.,
the underlying RMBS mezzanine tranches) is driven by a single common factorX,
which follows a standard normal distribution and is correlated with each Yi by ρ1 as:
Yi =√ρ1X+
√
and ξi for i = 1 to K represents the idiosyncratic term which follows a standard normal distribution and is independent of X.
Here we will assume additionally thatµ≥t2, which means the detachment point of the RMBS tranche is below the expected LGD of mortgages. This is nearly a
trivial assumption since it is almost universally true for all mezzanine tranches.
The expressions below will not be as simple if they are to be applied to senior
tranches where the assumption above does not hold, but the derivation will remain
the same.
The CDO portfolio that the RMBS tranche is put into can be regarded as a
“super-portfolio” as we defined previously. If we look at the default probability and
expected LGD of each RMBS within the super-portfolio, we can use the previous
results of equations (2.11) and (2.12):
P DRMBS =G(t1|X) =N ( 1 √ ρ(1−λ) [ N−1(p)−√1−ρN−1 ( t1 µ ) −√ρλX ]) , and ELGDRMBS=E[LRMBS|X]/P DRMBS = µ P DRMBS(t 2−t1) N2 ( N−1(p)−√ρλX √ 1−ρλ , N −1 ( t µ ) ; √ 1−ρ 1−ρλ ) t2 t1 .
Now we can see that the default probability and expected LGD of the RMBS
tranches are correlated through the common systematic factor X, although not in
a simplistic way. For example, the two are not necessarily monotonic with respect
underlying mortgages.
Heuristically, one can easily argue that when the thickness of a tranche
ap-proaches zero, the LGD of that tranche will approach 100%. This is stated in the
following proposition which applies to the unconditional case as well.
Proposition 0 (LGD of an infinitesimally thin tranche)
lim t2−t1↘0 ELGDRM BS = 1. Let F (s) ≡ E[LRMBS|Z =s;λ=ρ 1 ] as defined in equation (2.12). If K is
large enough, due to diversification, the qth quantile of the limiting loss can be
approximated by F(X1−q) as indicated by Theorem 3. Similar to the case for
RMBS tranches inCorollary 1, we can have the following
Proposition 1 (Stand-alone Capital of a CDO tranche)
The qth quantile of tranche loss or loss at the confidence level (LCL) is
F (X1−q)−T1 T2−T1 if T1 < F(X1−q)< T2 0 if F (X1−q)< T1 1 otherwise .
Because we assumed µ≥ t2, the probability of loss exceeding l can be written as: G(l) =N(F−1(l)). Following this technique, we can derive also the EL of the
Proposition 2 (Expected Loss of a CDO tranche)
For any CDO tranche [T1, T2], the expected loss can be computed by:
E[LCDO] = 1 T2−T1 T2 ∫ T1 dlG(l) = 1 T2−T1 T2 ∫ T1 dlN(F−1(l)) = µ (T2−T1) (t2−t1) [Q(η2)−Q(η1)], (2.14) where ηi = N −1(p)− √ρρ 1F−1(Ti) √ 1−ρρ1 i= 1,2 Q(η)≡N3 ( N−1(p), N−1 ( t µ ) , η; Σ) t2 t1 , Σ = 1 √1−ρ √1−ρρ1 √ 1−ρ 1 √ 1−ρ 1−ρρ1 √ 1−ρρ1 √ 1−ρ 1−ρρ1 1
and N3(a, b, c; Σ) denotes the trivariate normal distribution valued at (a, b, c) with
mean zero and covariance matrix Σ.
If a bank places the CDO tranche into their investment portfolio, which can be
regarded as a super-portfolio with the systematic factorZ, and it is correlated with
X as
X =√λ1Z +
√
1−λ1ζ
in which the correlation is denoted byλ1andζrepresents a standard normal random
Under this model assumption, the conditional probability of loss exceeding l
given the systematic factor can be written as:
G(l|Z) =N ( F−1(l)−√λ 1Z √ 1−λ1 ) .
Then the conditional expected loss can be written as:
E[LCDO|Z] = 1 T2 −T1 T2 ∫ T1 dlG(l|Z) = 1 T2−T1 T2 ∫ T1 dlN ( F−1(l)−√λ 1Z √ 1−λ1 ) = µ (T2−T1) (t2−t1) [ ˜ Q(η2)−Q˜(η1) ] , (2.15) where ˜ Q(η)≡N3 ( N−1(p)−√ρρ 1λ1Z √ 1−ρρ1λ1 , N−1 ( t µ ) , η; ˜Σ ) t2 t1 and ˜ Σ = 1 √ 1−ρ 1−ρρ1λ1 √ 1−ρρ1 1−ρρ1λ1 √ 1−ρ 1−ρρ1λ1 1 √ 1−ρ 1−ρρ1 √ 1−ρρ1 1−ρρ1λ1 √ 1−ρ 1−ρρ1 1 .
Thus, we have the following proposition:
Proposition 3 (Capital of a CDO tranche in the context of a super-portfolio)
The capital of the CDO tranche [T1, T2] within the super-portfolio can be
cal-culated as E[LCDO|Z 1−q
]
−E[LCDO] as defined in (2.14) and (2.15).
tranches are now derived from their underlying collateral characteristics rather than
directly specified as in LGDs of mortgages in the securitization model. Correlation
between the RMBS tranches ρ1 is also an additional parameter that we can set to
be much higher than AVC between mortgages.
What is still similar to the securitization model is thatρ1now has the same effect
that ρhad before. Higher correlation between RMBS tranches will favor low CDO
tranches and disfavor high CDO tranches. The correlation to the super-portfolio
still plays a similar role as in the securitization model. We will see more concrete
examples in the numerical section and get a better view of the parameters.
Note that our results above can be easily extended to the non-homogeneous
case. If we assume that the ith RMBS tranche’s PD, LGD, tranche limits and
weights in the CDO portfolio are pi, µi, ti1, ti2 and wi, respectively. We have the
following proposition:
Proposition 4 (Loss distribution of a non-homogeneous CDO)
If Gordy’s conditions (A-1) to (A-4) are satisfied, we have the same expected
loss and capital expressions as inProposition 1,2,3, except that F (X) has to be
rewritten as F (X) = K ∑ i=1 wiE [ LRMBSi |X] = K ∑ i=1 wiµi ti2−ti1 N2 ( N−1(pi)− √ ρλX √ 1−ρλ , N −1 ( t µi ) ; √ 1−ρ 1−ρλ ) ti2 ti1 .
2.3.3.3 Granularity Adjustment
While the number M of underlying mortgages in a RMBS is very large, the
number of underlying RMBS tranchesKin a CDO of RMBS may not be. Therefore,
a granularity adjustment becomes necessary. The loss of a coarse-grained CDO with
K RMBS tranches can be written as:
LCDOK =E[LCDO|X]+R(X) +O(K−2)
in whichR(X) stands for the first order adjustment. If we neglect terms of higher
orders than K−1 and let F (X) = E[LCDO|X]+R(X), the qth quantile of the CDO portfolio loss can still be approximated by F(X1−q), and the probability of
loss exceeding l can still be written as G(l) =N(F−1(l)). As a result, the
stand-alone capital of a coarse-grained CDO tranche will have a similar expression as in
Proposition 1 with the newly defined F(X).
Accordingly, to derive the expected loss of a coarse-grained CDO tranche, one
can follow the integration by parts as in (2.14):
T2 ∫ T1 dlN(F−1(l))= [ N(s)F (s)− ∫ ds n(s)F (s)] F−1(T 2) F−1(T 1) .
The computation will have to be carried out by numerical integration in this case.
One important thing to note here is that the above granularity adjustment for EL
may not work for equity tranches or some ‘junior’ mezzanine tranches that are just
above the equity tranche in the capital structure. Because when losses are small,
Now the only task left is for us to derive the expression of the granularity
adjust-ment term R(x). Let π(x) ≡ E[LRMBS|X =x] and ν(x) ≡ V ar[LRMBS|X =x]. Following Gordy (2003) and Wilde (2001),
R(x) = − 1 2Kπ′(x) [ v′(x)−v(x) ( π′′(x) π′(x) +x )] . (2.16) Let ˜ x≡ N −1(p)− √ρρ 1x √ 1−ρρ1 and ρ˜≡ 1−ρ 1−ρρ1. Equation (2.12) leads to π(x) = µ t2−t1 N2 ( ˜ x, N−1 ( t µ ) ;√˜ρ) t2 t1 . (2.17)
Then we can take the first and second derivatives:
π′(x) = µ t2−t1 (√ ρρ1 1−ρρ1 ) n(˜x)N (√ ˜ ρx˜−N−1(t/µ) √ 1−˜ρ ) t2 t1 (2.18) π′′(x) = µ t2−t1 ( ρρ1 1−ρρ1 ) n(˜x) · [ ˜ xN (√ ˜ ρx˜−N−1(t/µ) √ 1−˜ρ ) − √ ˜ ρ 1−˜ρ n (√ ˜ ρx˜−N−1(t/µ) √ 1−˜ρ )] t2 t1 (2.19) in which H(t)≡ √ 1 1−˜ρN −1 ( t µ ) .
The conditional loss variance and its derivative can be calculated as follows:
ν(x) = 2µ 2 (t2−t1) 2 N3 ( N−1 ( t µ ) ,0,x˜; ˆΣ) t2 t1 − 2t1 t2−t1 π(x)−π2(x) (2.20)
in which ˆ Σ = 1 −√1/2 √ρ˜ −√1/2 1 −√˜ρ/2 √ ˜ ρ −√˜ρ/2 1 , and ν′(x) = n(˜x)N2 (√ ˜ ρx˜−N−1(t/µ) √ 1−˜ρ , √ ˜ ρ 2−ρ˜x˜; √ 1−˜ρ 2−˜ρ ) t2 t1 · 2µ2 (t2−t1)2 (√ ρρ1 1−ρρ1 ) − 2t1 t2−t1 π′(x)−2π(x)π′(x). (2.21)
Plugging equations (2.17) - (2.21) into (2.16), we can obtain the granularity
adjust-ment term R(x).
Corollary 2 (Granularity Adjustment term for a coarse-grained CDO)
Using the above notations, the granularity adjustment term for a coarse-grained
CDO can be calculated as
R(x) = − 1 2Kπ′(x) [ v′(x)−v(x) ( π′′(x) π′(x) +x )] ,
where π′(x), π′′(x),v(x) and v′(x) are as defined in equations (2.17) - (2.21).
2.3.3.4 Numerical Results
The CDO we consider here in this section has five different tranches and is made
up of K = 30 RMBS mezzanine tranches. The RMBS tranches are homogeneous
Confidence level q = 99.9% is used, and the correlation between the CDO and the
super-portfolio is assumed to be 0.9. Results are also shown with a range of possible
correlation between RMBS tranches, from 0.5 to 0.8.
For comparison to our theoretical results, we also performed a ‘look-through’
simulation on the CDO, where loss distributions of RMBS tranches are generated
according to equation (2.7) and then aggregated for computation of EL and loss
quantile or loss at the confidence level (LCL) of the CDO. All simulation results
are based on 1,000,000 Monte Carlo samples.
The following columns are displayed in Table 2.2: LCL(sim) or LCL from
simu-lation, LCL(sa) or standalone LCL, LCL(adj) or LCL with granularity adjustment,
LCL(sp) or LCL in the context of a super-portfolio, EL(sim) or EL from
simula-tion, EL(grn) or EL from analytical solution assuming granular CDO portfolio, and
EL(adj) or EL with granularity adjustment.
It can be seen from the tables that across different RMBS tranche correlations,
the theoretical LCL values are fairly close to their simulation-based counterparts
for most of the tranches. After granularity adjustment, the two values agree even
more. The same can be said for expected loss except for the equity tranche, where
adjusted EL does not work as noted before.
As in the case of securitizations, the LCL values in the context of a
super-portfolio exhibit the phenomenon of transferring capital from junior tranches to
Table 2.2: EL and LCL with K = 30 Tranche
Limits(%) LCL(sim) LCL(sa) LCL(adj) LCL(sp) EL(sim) EL(grn) EL(adj)
RMBS Tranche Correlation = 0.5 0.0 - 6.0 100 100 100 99.2367 3.7666 3.9826 4.8402 6.0 - 7.0 100 100 100 94.9089 1.0023 0.7931 0.9916 7.0 - 15.0 100 100 100 74.2016 0.4605 0.3639 0.4551 15.0 - 50.0 17.0357 11.6144 16.8975 9.8743 0.0504 0.0387 0.0494 50.0 - 100.0 0 0 0 0.0017 0.0006 0.0005 0.0007 RMBS Tranche Correlation = 0.6 0.0 - 6.0 100 100 100 99.4750 3.2761 3.4898 3.9971 6.0 - 7.0 100 100 100 96.9279 1.1016 0.9679 1.1002 7.0 - 15.0 100 100 100 85.0848 0.5826 0.5123 0.5817 15.0 - 50.0 35.3093 30.0065 34.4230 21.9768 0.0944 0.0814 0.0935 50.0 - 100.0 0 0 0 0.0940 0.0036 0.0025 0.0031 RMBS Tranche Correlation = 0.7 0.0 - 6.0 100 100 100 99.5047 2.7826 2.9278 3.1624 6.0 - 7.0 100 100 100 97.5858 1.1390 1.0555 1.1313 7.0 - 15.0 100 100 100 90.2197 0.6766 0.6282 0.6729 15.0 - 50.0 58.8438 52.9004 56.4758 37.2015 0.1545 0.1397 0.1509 50.0 - 100.0 0 0 0 1.0598 0.0102 0.0088 0.0100 RMBS Tranche Correlation = 0.8 0.0 - 6.0 100 100 100 99.3644 2.2087 2.3116 2.3580 6.0 - 7.0 100 100 100 97.5508 1.0768 1.0489 1.0825 7.0 - 15.0 100 100 100 92.4092 0.7079 0.6933 0.7157 15.0 - 50.0 87.2661 82.5996 85.2957 52.7641 0.2154 0.2088 0.2168 50.0 - 100.0 0 0 0 5.1389 0.0269 0.0241 0.0258
Table 2.3: CDO Tranche 15% - 50% RMBS
Correlation 30 50 100 200 500 ∞
Loss at the 99.9% Confidence Level
0.5 16.8975 14.7842 13.1993 12.4069 11.9314 11.6144 0.6 34.4230 32.6564 31.3315 30.6690 30.2715 30.0065 0.7 56.4758 55.0456 53.9730 53.4367 53.1149 52.9004 0.8 85.2957 84.2172 83.4084 83.0040 82.7613 82.5996 Expected Loss 0.5 0.0494 0.0449 0.0417 0.0402 0.0393 0.0387 0.6 0.0935 0.0885 0.0849 0.0831 0.0821 0.0814 0.7 0.1509 0.1463 0.1430 0.1413 0.1403 0.1397 0.8 0.2168 0.2136 0.2112 0.2100 0.2093 0.2088
capital any more as in other standalone cases.
In Table 2.3, we focus on LCL and EL values for the (15%, 50%) CDO tranche
in the previous example with different level of granularity (K). It is straightforward
to observe that the granularity adjustm
